Generalization of the brauer theorem to matrix polynomials and matrix Laurent series

Given a square matrix A, Brauer’s theorem [Duke Math. J. 19 (1952), 75-91] shows how to modify one single eigenvalue of A via a rank-one perturbation, without changing any of the remaining eigenvalues. We reformulate Brauer’s theorem in functional form and provide extensions to matrix polynomials and to matrix Laurent series A(z) together with generalizations to shifting a set of eigenvalues. We provide conditions under which the modified function Ã(z) has a canonical factorization Ã(z) = Ũ(z) (Formula Present)(z-1) and we provide explicit expressions of the factors Ũ(z) and (Formula Present) (z). Similar conditions and expressions are given for the factorization of Ã(z-1). Some applications are discussed.